3. Consider a network (G, w). Let c ER, and let m: E(G) → R such that m(e) =w(e) + c for all e Є E(G).(a) Show that T is a minimum spanning tree of (G, w) if and only if it is a minimumspanning tree of (G,m).

Ans-3)(a) To show that T is a minimum spanning tree of (G, w) if and only if it is a minimum…

Answered: 3. Consider a network (G, w). Let c €…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(V, E) be a graph and c: E →→ R be an edge cost function. We call an edge e EE (1) necessary if e is contained in every minimum spanning tree; (2) useless if no minimum spanning tree contains e; (3) optional else. How many edges in the graph below are necessary? Let G H 5 2 2 1 2 3 2 2 1 3 2 5 CIT 6 2 2 5 2 3
5. Let G = (V, E) be a bipartite graph with partite sets X and Y. Suppose X = {x₁,x₂,...,xn} and Y = {y₁, y2,...,Ym}. Show that Σdc(x₁) = |E|. i=1 m (HInt: Think graph fairy). (Also, note that Σdc(y;) = |E| as well, but just prove the above). j=1
Let (u, v) be a minimum-weight edge in a connected graph G. Show that (u, v) belongs to some minimum spanning tree of G.
(3) Consider the weighted graph G: a b d g h Weights for G: f Edge Weight Edge Weight Edge Weight ab 2 cd 14 eg 12 ad 8 се 13 ei 12 ае 10 cf 16 fg 1 af 5 de 11 fh 15 bc 15 dg 6 fi 7 bd 9 dh 17 gh 19 bf 4 di 2 hi 20 ef 18 Use Kruskal's algorithm to find a minimal connector and state its weight.
Let {V₁,..., Vn} be a spanning set for an inner product space V. Suppose that x and y are such that (x, vk) = (y, vk) for all k. Prove that x = y. . Please prove this from definitions and do not refer to previous review, hw, or gw problems (you will lose points if you do so). If you use a theorem or property from the text (that has been proved), please refer to which one.
(a) (b) S = 5 12 (0·0·000} 1 Is S a spanning set for R¹? Justify your answer. 10 7 Is S linearly independent? Justify your answer.
Consider an undirected graph G = (V, E). An independent set is a subset I CV such that for any vertices i, j e I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let a(G) denote the size of the largest maximal independent set in G.
Which of the following are true statements? If {y, z} is linearly dependent, then {x, y, z} is 'linearly dependent for any X. Every set containing a spanning set is again a spanning set. If {x, y, z} is linearly independent, then {y, z} is linearly independent. If all of X1, X2,..., Xk are nonzero, then {X1, X2,...,xk} is linearly independent. If {x, y} is linearly independent, then {x, y, x+y} is linearly independent. If one of X1, X2,..., x is 0, then {x, y, z} is linearly dependent. If {x, y, z} is linearly independent, then ax+by+cz = 0 for some a, b and c in R.

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3. Consider a network (G, w). Let c € R, and let m: E(G) → R such that m(e) = w(e) + c for all e € E(G). (a) Show that T is a minimum spanning tree of (G, w) if and only if it is a minimum spanning tree of (G,m).